Bad alternator??

Hello all,

I'm trying to figure out if I have a bad alternator. For one, my battery symbol illuminated and then the electrical power started to get low then when I got home the battery completely died. I swapped out batteries from by brother's GT and the car started but I still got the battery light. I then disconnected the neg wire from the battery and the car immedietly died. Wouldn't this suggest a bad alt since the alt provies all electrical power once the car is on? Also, if you have a good alt, shouldn't you be able to disconnect the battery while the car is running and the car should continue to run right? I haven't hooked up a voltmeter yet but I have an autometer volt gauge and it reads around 12 while the car is running with my brother's battery. Any ideas?